Lets consider a ultra cold neutron gas in a gravitational potential. The known quasi-classical energy up to the classical turning point is
$$ E_n =\sqrt[3]{\frac{9m}{8}(\pi\hbar g (n-\frac{1}{4}))^2} $$ My question is that in this potential that has a vertical axis representing the potential $U(z)$ and $x$ horizontal axis representing the width of the well $z$, what would the potential be beyond the classical turning point?
Well, the form of the potential beyond the turning point depends on what's there.
When Nesvizhevsky et al did this at ILL, the lower surface was a neutron mirror, and the upper surface was an absorber. (Actually I can't access that paper at the moment, so perhaps I'm misremembering; I'll correct if necessary.)
A good model for the mirror is a uniform Fermi pseudopotential of about 100 neV inside the mirror, and zero out; a good model for an absorber is a psuedopotential with a real part near zero, and a nonzero complex part which causes the neutrons to disappear as time evolves.
Note that the Nesvizhevsky-type experiments were with a collimated beam of UCNs, so that the part of their kinetic energy in the vertical direction was quite small compared to the "usual" UCN energy of 100 neV. A "gas" suggests something more like the PNPI GraviTrap, which had no lid, and for which I don't believe there's been any evidence of quantized gravitational states.
